Could someone please tell me if this sentence is correct?
"Es sagt, daß das verboten sei."
The word 'sei' is a conjugation for 'sein' using the mode Konjunktiv I (I think it's called the subjunctive in English), which is often used for reported speech (and similar situations). This is commonly used in newspapers, for example.
A good description of Konjunktiv I: http://www.lsa.umich.edu/german/hmr/Grammatik/Konjunktiv_I.html
If you use 'ist' instead, then you are using the indicative mode, which is for stating facts. It is a lot more common to hear the indicative in spoken German.
From a normative point of view "Es sagt, daß das verboten sei." is grammatically correct. The rule is that you must use Konjunktiv, if otherwise an indirect speech would not be readily identifiable:
Er sagt: "Das ist verboten."
Indirect speech with subordinate clause:
Er sagt, dass das verboten ist. Er sagt, dass das verboten sei.
Indirect speech only with Konjunktiv:
Er sagt, das sei verboten. *Er sagt, das ist verboten. **wrong!**
From a descriptive point of view, most native speakers of German never use the Konjunktiv. They don't know how to use it, and avoid it. Therefore, in spoken German, you will hear people say: "Er sagt, das ist verboten." (more common, less halting), or: "Er sagt, dass das verboten ist." (Please note that native speakers indeed do use the grammatically "wrong" form!)
In spoken German the Konjunktiv is perceived as elevated, even stilted. For the most part, only people whose business is language (writers, journalists, politicians etc.) use it to express that they quote the opinion of another person.
Even in written language, the Konjunktiv is not common, but since written language is more normative, you'll find that more native speakers try to apply it. Here you would find "sei" more often, although most readers would not see "ist" as faulty, unless some German teacher pointed the mistake out to them.
What you should use would depend on the context. I have known non-native speakers who speak German more correctly than any native speaker would, and it often gives them an air of otherworldliness. If you speak, speak like the natives do. If you write, use the Konjunktiv.
It is correct.
As mentioned by other answers in everyday speech people do not use it that often. Either the simply put direct speech
Er hat gesagt: "Das ist verboten".
or they just ignore Konjunktiv
Er hat gesagt, dass das verboten ist.
Anyhow if one uses it, it emphasizes that one is only reproducing the statement of somebody else, thus dissociating oneself from what was said.